| Summary: | 碩士 === 國立中興大學 === 應用數學系所 === 106 === A sign pattern (or sign pattern matrix) is a matrix whose entries come from the set ${-,0,+}$. For a real matrix $A$, $sgn(A)$ is the sign pattern of $A$ whose entries are the signs of the corresponding entries in $A$. If $mathcal{A}$ is a sign pattern, the sign pattern class of $mathcal{A}$ is denoted by Q($mathcal{A}$), which is the set of all real matrices $A$ with $sgn(A)=mathcal{A}$. If $mathcal{P}$ is a property referring to a real matrix, then a sign pattern $mathcal{A}$ $requires$ $mathcal{P}$ if every real matrix in Q($mathcal{A}$) has property $mathcal{P}$. A positive matrix, $A>0$, is a matrix all of whose entries are positive real numbers. A square real matrix $A$ is said to be algebraically positive if there exists a real polynomial $p$ such that $p(A)$ is a positive matrix.
In this thesis, we study the sign patterns that require algebraic positivity. We give a characterization of $4 imes 4$ symmetric tridiagonal sign patterns that require algebraic positivity.
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